Multiple-Edge-Fault-Tolerant Approximate Shortest-Path Trees

Abstract

Let G be an n-node and m-edge positively real-weighted undirected graph. For any given integer $f \ge 1$, we study the problem of designing a sparse f-edge-fault-tolerant (f-EFT) $\sigma$-approximate single-source shortest-path tree ($\sigma$-ASPT), namely a subgraph of G having as few edges as possible and which, following the failure of a set F of at most f edges in G, contains paths from a fixed source that are stretched by a factor of at most $\sigma$. To this respect, we provide an algorithm that efficiently computes an f-EFT $(2|F|+1)$-ASPT of size O(fn). Our structure improves on a previous related construction designed for unweighted graphs, having the same size but guaranteeing a larger stretch factor of $3(f+1)$, plus an additive term of $(f+1) \log n$. Then, we show how to convert our structure into an efficient f-EFT single-source distance oracle, that can be built in $O(f m\, \alpha (m,n)+fn \log ^3 n)$ time, has size $O(fn \log ^2 n)$, and in $O(|F|^2 \log ^2 n)$ time is able to report a $(2|F|+1)$-approximate distance from the source to any node in $G-F$. Moreover, our oracle can return a corresponding approximate path in the same amount of time plus the path’s size. The oracle is obtained by tackling another fundamental problem, namely that of updating a minimum spanning forest (MSF) of G following a batch of k simultaneous modification (i.e., edge insertions, deletions and weight changes). For this problem, we build in $O(m \log ^3 n)$ time an oracle of size $O(m \log ^2 n)$, that reports in $O(k^2 \log ^2 n)$ time the (at most 2k) edges either exiting from or entering into the MSF. Finally, for any integer $k \ge 1$, we complement all our results with a lower bound of $\Omega \left( n^{1+\frac{1}{k}}\right)$ to the size of any f-EFT $\sigma$-ASPT with $f \ge \log n$ and $\sigma < \frac{3k+1}{k+1}$, that holds if the Erdős’ girth conjecture is true.